Quantitative information is fundamental for biological systems. 
For example, behavior in most biochemical reactions
is highly dependent on the presence of a certain amount of the substances involved. 
Very often, information is \emph{partial} as obtaining exact values for parameterizing models is difficult.
Unpredictable behavior is thus an inherent condition of the biologic phenomena, and one counts with \emph{partial behavioral information} for describing system interactions.
This partial information not only ignores elements on \emph{how} reactions occur (e.g. what components actually interact), but also on \emph{when} such reactions commonly happen (e.g. the relative speeds of the interacting components).

While the notion of partial quantitative information is central to \ccp via constraints, 
partial behavioral information is actually the novelty of \ntcc\ via non-deterministic and asynchronous operators. 
Our teams have already explored these advantages by analyzing mechanisms for cellular transport 
and genetic regulatory networks  \cite{gutierrez07,Arbelaez07}.

A drawback of these models is their lack of explicit quantitative information.
As hinted at above, a fundamental feature of any model of biological systems is the capability of exploiting any available
quantitative information.
In biological systems this is often represented as \emph{stochastic behavior}.   
One then has a set of %(biochemical) 
reactions  each endowed with a  rate  representing  
their  propensity  or speed.  When considering their execution,
%In order to simulate the execution of 
%several reactions, 
a race between them takes place and the fastest action is executed.  
%In practice, this approach for the simulation of chemical reactions relies on Gillespie's algorithm  \cite{Gillespie77}. 


%\todo{Please add another sentence explaining why...}
We have taken initial steps on the inclusion
%In FORCES, we have taken initial steps on the inclusion 
of stochastic information into an explicitly timed concurrent constraint process language \cite{ArandaPRV08}. We defined stochastic events in terms of the time units defined by the language: this provides great flexibility for modeling and allows for a clean semantics. Most importantly, by considering stochastic information and adhering to explicit discrete time, it is possible to reason about processes using quantitative logics (both discrete and continuous), while retaining the simplicity of calculi such as \ntcc\ for deriving qualitative reasoning techniques (such as denotational semantics and proof systems). 
We plan to consolidate the framework outlined 
in \cite{ArandaPRV08}, and to apply it
to study systems such as the modeled
%it to model the cycle of Rho GTP-binding proteins in the context of phagocytosis, as proposed 
in \cite{CardelliGK08}. \\


%In \cite{beauxis08-tech} and \cite{beauxis08} we studied an extension of CCP with probabilistic executions. We developed a mathematical framework for defining the meaning of an infinite probabilistic execution. We used a topological notion of probability called valuations, and we gave the conditions under which the limit of an infinite execution exists and enjoys the expected properties. Using this result, we defined a denotational semantics in which closure operators on constraints are replaced by linear closure operators on vector spaces. Using this language we proposed a model of Crowds, a probabilistic anonymous protocol. We plan to use the denotational semantics in  \cite{beauxis08-tech} to analyze anonymity properties of this kind of protocols.
